Diffusion Approximation for Open State-Dependent Queueing Networks in the Heavy Traffic Situation

Ergodic Rate Control Problem for Single Class Queueing Networks

“…We note that conditions in [24] are somewhat different from those considered here. The paper [24] imposes a restrictive form of state dependence on the rates λ n , µ n ; namely it requires for x = (x 1 , . .…”

“…Remarks on the Proof. The proof follows that of Theorem 1 of [24]. We note that conditions in [24] are somewhat different from those considered here.…”

“…Many works that study diffusion approximations for queueing systems (eg. [14], [18]) formulate the underlying heavy traffic condition in terms of rates that are O (1) and then consider scaled processes where time is scaled up by a factor of n while space is scaled down by a factor of √ n. In the scaling considered here (see (2.7)), which is adapted from [24] (see also [20]), the time parameter is left unchanged and the factor of n is absorbed in the arrival and service rates. A simple example of rate functions satisfying (ii)-(vii) is given…”

“…However, [24] imposes weaker convergence requirements on these rates than Assumption 2.1 (v), (vi) of the current paper (see A2, A4 in [24]). Also, the limit process in [24] is expressed in terms of a [K + K(K + 1)] dimensional Brownian motion and the diffusion coefficient matrix Σ. One finds that the proofs in [24] can be carried over to the current setting with only minor modifications.…”

“…The heavy traffic assumption says that traffic intensity at each station is of the order 1 − O(1/n 1/2 ). In an uncontrolled setting, diffusion approximations of suitably scaled queue length processes for such networks have been studied in [24] (see also [20]). The limit stochastic process is a reflected diffusion in the nonnegative orthant.…”